Controllable generation and propagation of ultraslow optical solitons via parameters

management in a five-level hyper inverted-Y atomic system

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IOP PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 (7pp) doi:10.1088/0953-4075/42/22/225405

Controllable generation and propagationof ultraslow optical solitons viaparameters management in a five-levelhyper inverted-Y atomic systemLiu-Gang Si, Xin-You Lu, Jia-Hua Li, Xiangying Hao and Meng Wang

Wuhan National Laboratory for Optoelectronics, School of Physics, Huazhong University of Science andTechnology, Wuhan 430074, People’s Republic of China

E-mail: siliugang@163.com

Received 26 July 2009, in final form 29 September 2009Published 30 October 2009Online at stacks.iop.org/JPhysB/42/225405

AbstractThe dynamics of generation and propagation of ultraslow optical solitons in alifetime-broadened five-level hyper inverted-Y atomic system are investigated. Due to thenovel absorption and dispersion properties of this system which provide the necessaryingredients for making the probe field propagate nearly transparent in three regimes, thegeneration of bright or dark optical solitons can be controlled with parameters management byactively manipulating the dispersion, the nonlinearity and the gain (absorption coefficient) viaadjusting the corresponding one-, two- and three-photon detunings and the Rabi frequencies.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

The interaction between an electromagnetic field and anonlinear medium [1–10] may result in many interestingphenomena, one intriguing example is the optical soliton[11–20] which describes the particle-like properties of anoptical pulse envelope in nonlinear systems. It formsby a proper balance between the nonlinear effects andthe dispersion properties of the medium under opticalexcitation. Until now, most optical solitons have beenproduced in passive optical media, such as optical fibres[1–3] which lack distinctive energy levels and strong nonlineareffects. Therefore, intense electromagnetic fields, substantialpropagation distance and far-off resonance excitation schemesare generally employed in order to avoid unmanageable opticalfield attenuation and distortion. In this case, the opticalsolitons generally have a speed very close to the velocity ofelectromagnetic wave in free space.

Recently, with the observation of electromagneticallyinduced transparency (EIT) [21–24] and large Kerrnonlinearity [25–27], there has been a significant surge ofresearch activities on wave propagation in highly resonantmedia. Many optical processes including optical bistability

[28, 29], quantum entanglement [30–32] and highly efficientfour-wave mixing (FWM) [33–36] in such highly resonantmedia have been reported. In particular, Wu and Deng havegeneralized their schemes of highly resonant media based onRaman excitation for the generation of a class of opticalsolitons, named ultraslow optical solitons [11, 12]. Theyshowed that the formation of this fascinating shape-preservingpulse wave with ultraslow group velocities in a cold atomicsystem is a fairly universal phenomenon. Subsequently, manyschemes in atomic systems have been proposed with the aimof producing various other types of optical solitons [37–40].These fruitful research findings have expanded our knowledgeof the field of resonant nonlinear optics in coherent media[41, 42].

In the present paper, we will explore the possibility ofthe generation and propagation of ultraslow optical solitonsin a lifetime-broadened five-level hyper inverted-Y atomicsystem. A similar system has been used to study the largeenhancement of FWM [34] and the generation of coherentlight [43]. Here we will demonstrate that this five-levelhyper inverted-Y atomic system can provide all the necessaryingredients for generating both dark and bright optical solitonswith ultraslow group velocities. More importantly, the system

0953-4075/09/225405+07$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

p

3

a

4

10

2

b

a

c

p

b

c

Figure 1. The energy-level diagram and excitation scheme of alifetime-broadened five-level hyper inverted-Y atomic systeminteracting with three strong cw control fields �a,b,c, and a weak,pulsed probe field �p to generate ultraslow optical solitons. �p isthe one-photon detuning, �a and �b are two separate two-photondetunings and �c represents the three-photon detuning for relevanttransitions.

has novel absorption and dispersion properties for making theprobe field propagate nearly transparently in three regimes,which can be used to control the generation of bright ordark optical solitons with parameters management [44, 45]by actively manipulating the dispersion, the nonlinearity andthe absorption coefficient via adjusting the correspondingone-, two- and three-photon detunings, as well as the Rabifrequencies.

The rest of our paper consists of three sections as follows.Section 2 aims to describe the model. We derive thecorresponding Hamiltonian, the atomic equations of motionand the governing equation for slowly varying envelopeof a pulsed probe field. We also discuss the linearizeddynamics of the system in this section. In section 3, afterconsidering the nonlinear effects in the propagation of thepulsed probe field, we obtain a nonlinear envelope equationgoverning the time evolution of the probe field envelope andshow that optical solitons can propagate through the atomicsystem with ultraslow group velocities. Besides, we brieflydiscuss the controllability of the generated solitons beingbright or dark with parameters management. With conclusionsand discussions being made in section 4, we end ourpaper.

2. Theoretical model

We consider a lifetime-broadened five-state atomic system ina hyper inverted-Y configuration as depicted in figure 1, wherethe transition |1〉 ↔ |2〉 (|2〉 ↔ |3〉, |3〉 ↔ |4〉) is coupledby a strong continuous wave (cw) control field with wavevector �ka (�kb, �kc), optical frequency ωa (ωb, ωc) and Rabifrequency �a (�b, �c), and the transition |0〉 ↔ |2〉 is drivenby a weak pulsed probe field with wave vector �kp, opticalfrequency ωp and Rabi frequency �p. Inspecting the five-levelhyper inverted-Y system (see figure 1), we find that this systemconsists of two sub-systems, i.e. one �-type (|0〉 ↔ |2〉 ↔ |1〉)and another four-state cascade-type (|0〉 ↔ |2〉 ↔ |3〉 ↔ |4〉).In order to cancel Doppler broadening and reduce interatomic

collisions, the atoms are trapped in a cell at sufficiently lowtemperature.

Under the electric-dipole and the rotating-waveapproximations, the Hamiltonian describing the system shownin figure 1 is given by

H

h=

4∑j=1

ωj0|j 〉〈j | − (�p e−iωpt+i�kp ·�r |2〉〈0|

+ �a e−iωat+i�ka ·�r |2〉〈1| + �b e−iωbt+i�kb ·�r |3〉〈2|+ �c e−iωct+i�kc ·�r |4〉〈3| + H.c.), (1)

where ωj0 = (εj − ε0)/h is the resonant transition frequencywith (ε0) εj (j = 1, 2, 3, 4) being the energy of the atomicstate (|0〉) |j 〉 and 2�p,a,b,c = (�μ20,21,32,43 · �ep,a,b,c)Ep,a,b,c/h

denote the Rabi frequencies for the respective transitions with�μlm denoting the dipole moment for the transition betweenlevels |l〉 and |m〉, and �ep,a,b,c being the unit vector inthe corresponding polarization direction. In the interactionpicture, by appropriately choosing the free Hamiltonian(H0/h = (ωp −ωa)|1〉〈1| +ωp|2〉〈2| + (ωp +ωb)|3〉〈3| + (ωp +ωc + ωb)|4〉〈4|), the interaction Hamiltonian can be written as

Hint /h = −�a|1〉〈1| − �p|2〉〈2| − �b|3〉〈3| − �c|4〉〈4|− (�p ei�kp ·�r |2〉〈0| + �a ei�ka ·�r |2〉〈1|+ �b ei�kb ·�r |3〉〈2| + �c ei�kc ·�r |4〉〈3| + H.c.), (2)

where �p = ωp − ω20 is the one-photon detuning, �a =ωp −ωa −ω10 and �b = ωp + ωb −ω30 are two separate two-photon detunings, respectively, and �c = ωp + ωb + ωc − ω30

represents the three-photon detuning.In the interaction picture, defining the atomic state

as |�〉 = A0(t)|0〉 + A1(t) ei(�kp−�ka)·�r |1〉 + A2(t) ei�kp ·�r |2〉 +A3(t) ei(�kp+�kb)·�r |3〉 + A4(t) ei(�kp+�kb+�kc)·�r |4〉, we readily obtain,from Schrodinger and Maxwell’s equations, the equationsgoverning the motion of atomic state amplitudes and the waveequation for slowly varying envelope of the time-dependentprobe field

∂A1

∂t= i(�a + iγ1)A1 + i�a

∗A2, (3a)

∂A2

∂t= i(�p + iγ2)A2 + i�aA1 + i�b

∗A3 + i�pA0, (3b)

∂A3

∂t= i(�b + iγ3)A3 + i�bA2 + i�c

∗A4, (3c)

∂A4

∂t= i(�c + iγ4)A4 + i�cA3, (3d)

∂�p

∂z+

1

c

∂�p

∂t= iκ20A2A0

∗, (3e)

|A0|2 = 1 − |A1|2 − |A2|2 − |A3|2 − |A4|2, (3f )

where γj is the decay rate of state |j 〉 (j = 1, 2, 3, 4) andκ20 = Nωp| �μ20 · �ep|2/(2hε0c) with N and ε0 being theconcentration and vacuum dielectric constant, respectively.

To provide a clear picture of the interplay between thegroup-velocity dispersion and nonlinear effects of the atomicsystem interacting with three strong cw optical fields and aweak pulsed probe field, we first investigate the linear property

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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

of the system. This requires a perturbation treatment of thesystem response to the first order of the probe field �p whilekeeping all orders of the control fields �a,b,c. In the nextsection, the higher order of �p that can greatly enhance thenonlinear effects in the propagation of the probe field will beconsidered, which may balance the dispersion effects and leadto the formation of ultraslow optical solitons.

We assume that all atoms are initially populated in theground state as shown in figure 1 and the Rabi frequency �p

of the probe field is a small quantity compared with those of theother three control fields. In this case the nondepleted groundstate approximation (A0(t) ≈ 1) can be applied and we canmake the asymptotic expansion Aj = ∑

k A(k)j , where A

(k)j is

the kth-order part of Aj in terms of �p. Within the followingadiabatic framework, it can be shown that A

(0)j = δj0 and

A(1)0 = 0. Taking Fourier transform of equations (3a)–(3e)

and keeping up to the first order of �p, we have

(ω + �a + iγ1)β(1)1 + �a

∗β(1)2 = 0, (4a)

(ω + �p + iγ2)β(1)2 + �aβ

(1)1 + �b

∗β(1)3 = −�p, (4b)

(ω + �b + iγ3)β(1)3 + �bβ

(1)2 + �c

∗β(1)4 = 0, (4c)

(ω + �c + iγ4)β(1)4 + �cβ

(1)3 = 0, (4d)

∂�p

∂z− i

ω

c�p = iκ20β

(1)2 , (4e)

where β(1)j and �p are, respectively, the Fourier transforms of

A(1)j (j = 1, 2, 3, 4) and �p, and ω is the Fourier variable.

Equations (4a)–(4d) can be solved straightforwardly, thatis

β(1)1 = −S2(ω)�a

∗

S(ω)�p, (5a)

β(1)2 = S2(ω)(ω + �a + iγ1)

S(ω)�p, (5b)

β(1)3 = �b(ω + �a + iγ1)(ω + �c + iγ4)

S(ω)�p, (5c)

β(1)4 = −�b�c(ω + �a + iγ1)

S(ω)�p, (5d)

with S(ω) = |�a|2|�c|2 − |�a|2(ω + �b + iγ3)(ω + �c +iγ4)−|�b|2(ω + �c + iγ4)(ω + �a + iγ1)−|�c|2(ω + �a + iγ1)

(ω +�p + iγ2)+ (ω +�a + iγ1)(ω +�p + iγ2)(ω +�b + iγ3)(ω +�c + iγ4) and S2(ω) = |�c|2 − (ω + �b + iγ3)(ω + �c + iγ4).

Substituting equation (5b) into equation (4e) and makinguse of the initial condition of the probe field �p(0, ω), i.e. theFourier transform of the probe field at the entrance z = 0, wereadily obtain

�p(z, ω) = �p(0, ω)exp[izK(ω)], (6)

where

K(ω) = ω

c+ κ20

S2(ω)(ω + �a + iγ1)

S(ω)

= K0 + K1ω + K2ω2 + O(ω3) (7)

with

K0 = β + iα/2 = κ20d1S2/S, (8a)

K1 = 1/Vg

= 1

c+

κ20

S2{d1B1S2 + S[S2 − d1(d3 + d4)]}, (8b)

K2 = 2κ20

S3[d1SB2S2 + d1B1

2S2 + SB1S2

−SB1d1(d3 + d4) − (d1 + d3 + d4)S2], (8c)

d1 = �a + iγ1, d2 = �p + iγ2,

d3 = �b + iγ3, d4 = �c + iγ4, (8d)

S = �a|2|�c|2 − |�a|2d3d4

− |�b|2d4d1 − |�c|2d1d2 + d1d2d3d4, (8e)

S2 = |�c|2 − d3d4, (8f )

B1 = �a|2(d3 + d4) + |�b|2(d4 + d1) + |�c|2(d1 + d2)

− d1d2d3 − d2d3d4 − d3d4d1 − d4d1d2, (8g)

B2 = �a|2 + |�b|2 + |�c|2− d1d2 − d1d3 − d1d4 − d2d3 − d2d4 − d3d4. (8h)

Expressions (8a)–(8c) have a clear physical interpretation.The imaginary part of K0 presents the absorption coefficientα, while the corresponding real part β is the phase shift perunit length of the probe field. Thus, Vg = 1/K1 gives thegroup velocity of optical solitons evolved from the weak probefield, and K2 represents the group-velocity dispersion thatcontributes to the probe pulse’s shape change and additionalloss of probe field intensity.

To illustrate the linearized dynamics of the system, weplot in figures 2–5 the absorption (solid curve) and dispersion(dashed curve) spectra of the probe field �p via the imaginaryand real part of K(ω) for different cases. To bring about thispurpose, we take κ20 = 60, γ1 = 0.0001, γ3 = 0.1357, γ4 =0.0143. These parameters along with the Rabi frequencies andthe detunings used in the following are all scaled by γ2, whichis in the order of MHz for rubidium or sodium atoms. Firstof all, we see from figures 2–5 that three EIT windows alwaysoccur as long as three control fields �a,b,c are on. In figures2(a)–(d), the modifications of the absorption and dispersionspectra of the probe field by three control fields �a,b,c arepresented under the resonance conditions �a = �b = �c =�p = 0. Then we adjust the values of the Rabi frequencies ofthree control fields �a,b,c to equal one another and investigatethe impact of one-photon detuning �p, two-photon detunings�a and �b, and three-photon detuning �c on the absorptionand dispersion spectra of the probe field, which are shownin figures 3–5. These results show that the absorption anddispersion are either symmetric or asymmetric and their shapesdepend critically on the system parameters. Furthermore,this five-level hyper inverted-Y atomic system provides morepreferences to make the probe field propagate transparentlywith almost no absorption and may lead to the formation ofultraslow optical solitons. Then, these novel absorption and

3

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(a)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(b)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(c)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(d)

Figure 2. The absorption Im[K(ω)] (solid curve) and dispersion spectra Re[K(ω)] (dashed curve) of the probe field versus dimensionlessdetuning ω in a five-level hyper inverted-Y atomic system with parameters κ20 = 60, γ1 = 0.0001, γ3 = 0.1357, γ4 = 0.0143,�a = �b = �c = �p = 0 and (a) �a = �b = �c = 5, (b) �a = 2, �b = �c = 5, (c) �b = 2, �a = �c = 5, (d) �c = 2, �a = �b = 5.All parameters are in units of γ2.

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(a)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(b)

Figure 3. As figure 2 but with �a = �b = �c = 2, (a) �a = −2, �b = �c = �p = 2, (b) �c = −2, �a = �b = �p = 2.

dispersion properties of this system may be used to control thegeneration of bright or dark optical solitons with parametersmanagement by actively manipulating the dispersion, thenonlinearity and the absorption coefficient via adjusting thecorresponding one-, two- and three-photon detunings, as wellas the Rabi frequencies.

3. Nonlinear dynamics and solitons

This section is presented to investigate the nonlinear evolutionof the weak probe field. In the following we will demonstratethat the high order of �p greatly enhances the nonlineareffects in the propagation of the probe field, which balance

4

J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(a)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(b)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(c)

Figure 4. As figure 3 but with (a) �a = �p = −2, �b = �c = 2, (b) �b = �p = −2, �a = �c = 2, (c) �b = �c = −2, �a = �p = 2.

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(a)

−10 −5 0 5 10−30

−20

−10

0

10

20

30

40

50

60

Detuning (ω/γ2)

Abs

orpt

ion

and

Dis

pers

ion

(b)

Figure 5. As figure 3 but with (a) �a = �b = �c = −2, �p = 2, (b) �b = �c = �p = −2, �a = 2.

the group-velocity dispersion effects and lead to the possibleshape-preserving propagation of the probe field in the atomicsystem.

It is noteworthy that equation (6) is obtained in the linearregime of the system under adiabatic approximations, ignoringhigher order of the Rabi frequency �p of the weak probe field.In order to preserve the shape of the probe field, the highorder of �p which greatly enhances the nonlinear effects inthe propagation of the probe field must be considered on theright-hand side of equation (3e). Then A2A0

∗ [11, 12] can be

expressed asA2A0

∗ ≈ A(1)2

[A

(0)0

]∗+ A

(3)2

[A

(0)0

]∗+ A

(1)2

[A

(2)0

]∗

= A(1)2

[A

(0)0

]∗ − A(1)2

[∣∣A(1)1

∣∣2+

∣∣A(1)2

∣∣2+

∣∣A(1)3

∣∣2+

∣∣A(1)4

∣∣2],

(9)where solution of A2A0

∗ is retained to third-order part andA

(1)j (j = 1, 2, 3, 4) can be obtained from equations (5a)–(5d)

as follows:

A(1)1 = −S2�a

∗

S�p, A

(1)2 = d1S2

S�p, (10a)

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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

A(1)3 = d1d4�b

S�p, A

(1)4 = −d1�b�c

S�p. (10b)

With the help of equations (9) which contain the linearpolarization terms and the lowest order nonlinear polarizationterms, we seek the probe wave solution of the following form:�p(z, t) = �P(z, t)exp[izK0] [1, 11, 12], and substitute it intoequations (3a)–(3e) to obtain the nonlinear wave equation ofthe slowly varying envelope �P(z, t):

i

(∂

∂z+

1

Vg

∂

∂t

)�P − K2

∂2

∂t2�P = W e−αz|�P |2�P , (11)

with W = κ20d1S2[|S2|2(|�a|2 + |d1|2) + |�a|2|d1|2(|�c|2 +|d4|2)]/(S|S|2). Defining ξ = z and η = t − z/Vg , accordingto ∂/∂z ∼ ∂/∂ξ − ∂/(Vg∂η) and ∂/∂t ∼ ∂/∂η, equation (11)can be rewritten as

i∂

∂ξ�P − K2

∂2

∂η2�P = W e−αξ |�P |2�P , (12)

where K2 describes group velocity dispersion, W characterizesthe self-phase modulation effect and α is the absorbtioncoefficient of the probe field.

The nonlinear evolution equation (12) for �P(z, t) hascomplex coefficients and hence generally does not allowsoliton solutions. However, as we show below, the absorptionof the probe field can be largely suppressed by adjustingthe intensities of three control fields and the one-, two- andthree-photon detunings, and a reasonable and realistic set ofparameters for the present system can be actively adjusted sothat K2 = K2r + iK2i ≈ K2r and W = Wr + iWi ≈ Wr withK2r K2i and Wr Wi . In this case equation (12) can bereduced to the standard nonlinear Schrodinger equation

i∂

∂ξ�P − K2r

∂2

∂η2�P = Wr |�P |2�P , (13)

which has dark and bright soliton solutions [1–3], includingN-solitons for bright solitons [3]. The sign of product K2r ·Wr

determines that the solutions of equation (13) are bright ordark solitons. If the parameters of the system are chosen tosatisfy the condition K2r ·Wr < 0, then we can obtain the darksoliton solution in the following form [3]:

�P = �P0tanh(η/τ)exp[−iWr |�P0|2ξ ], (14)

while if the product K2r · Wr is positive under appropriateparameter conditions, equation (13) supports the bright N-soliton solutions. Then the fundamental bright soliton forN = 1 and the bright two-soliton (N = 2) of equation (13)are, respectively, given by

�P = �P0sech(η/τ)exp[−iWr |�P0|2ξ/2], (15)

�P = �P0

×4[cosh(3η/τ)+3exp(−8iK2r ξ/τ 2)cosh(η/τ)]exp(−iK2r ξ/τ 2)

cosh(4η/τ)+4cosh(2η/τ)+3cos(8K2r ξ/τ 2).

(16)

In equations (14)–(16), tanh(η/τ), sech(η/τ) andcosh(η/τ) are, respectively, the hyperbolic tangent, secant andcosine functions. τ is the pulse length of the probe field at theentrance of the medium, and the amplitude �P0 is determined

by the constraint relation |�P0τ |2 = 2|K2r/Wr |, whichdenotes the balance between the group-velocity dispersion andnonlinearity effects in our system.

Now we consider a practical atomic system that canbe realized by a typical alkali atomic vapour at sufficientlylow temperature to generate ultraslow dark or bright opticalsolitons. An alkali atom for the proposed system is 87Rb withthe designated states chosen as follows [43]: 5S1/2, F = 1as |0〉, 5S1/2, F = 2 as |1〉, 5P1/2 as |2〉, 5D3/2 as |3〉 andnP3/2 (n > 10) as |4〉. Thus, the decay rates of excited states|1〉, |2〉, |3〉, and |4〉 are, respectively, �1 = 2γ1 11.2 kHz,�2 = 2γ2 11.2 MHz, �3 = 2γ3 1.52 MHz and�4 = 2γ4 0.16 MHz.

We first consider the case of ultraslow dark opticalsolitons. According to the analysis above, to makeequation (13) have solutions describing dark solitons, thecorrelative coefficients have to satisfy α ≈ 0, K2r K2i ,Wr Wi , and K2r · Wr < 0. To attain this purpose, wetake κ20 ≈ 336 MHz −1, �a ≈ �b ≈ �c ≈ 112 MHz,�a ≈ �b ≈ �c ≈ 8.4 MHz and �p ≈ −560 MHz,then α ≈ 0.0011 cm−1, Vg/c ≈ 2.3 × 10−3, K2 ≈(−8.604 + 0.028i) × 10−16 s2 cm−1, and W ≈ (7.065 +0.023i) × 10−18 s2 cm−1. With these quantities, the standardnonlinear Schrodinger equation (13) with K2r ·Wr < 0 is wellcharacterized and the formation of dark solitons occurs. Thus,we have demonstrated the existence of dark optical solitonswith ultraslow propagation velocities compared with c in thisfive-level hyper inverted-Y atomic system.

Inspecting the group velocity dispersion coefficient K2

and the self-phase modulation coefficient W , we find that thesigns of them are dependent on the one-, two- and three-photon detunings, as well as the Rabi frequencies of threecontrol fields, which gives rich dynamics and characteristicsoliton switching (dark ⇐⇒ bright). More interestingly, ourscheme provides three regimes where the probe field �p canwork with a nearly negligible absorption, which can be seenfrom figures 2–5. Thus, we may provide more practicalopportunities to implement soliton production and propagationin this system thanks to the activity of manipulating thedispersion, the nonlinearity and absorption coefficient byadjusting the corresponding one-, two- and three-photondetunings and the Rabi frequencies. This is different from fibreoptic telecommunications, where the dispersion managementis introduced by applying a long fibre link consisting ofperiodically alternating segments of fibres with oppositesigns of the group-velocity dispersions [44]. It is worthmentioning that the scheme in a nonlinear optical fibre [46]is proposed to investigate the effective transformation of adark soliton into a bright one, where two waves involved inthis system are co-propagational with one wave being at theanomalous dispersion while the other one is at the normaldispersion. Here, with the existence of optical bright anddark solitons in the five-level hyper inverted-Y atomic system,we can easily achieve the transformation between dark andbright solitons by manipulating flexibly the correspondingdetuning and intensity conditions of input optical fields. Asa simple example, we only adiabatically adjust the two-photon detuning �a ≈ −0.5 MHz without changing other

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J. Phys. B: At. Mol. Opt. Phys. 42 (2009) 225405 L-G Si et al

parameters given above, then we have α ≈ 0.000 32 cm−1,Vg/c ≈ 1.2×10−3, K2 ≈ (−2.527+0.030i)×10−15 s2 cm−1,and W ≈ (−1.141 + 0.013i) × 10−18 s2 cm−1. With thesequantities, the standard nonlinear Schrodinger equation (13)with K2r · Wr > 0 is well characterized and the existenceof ultraslow bright optical solitons in this system has beendemonstrated.

4. Conclusions

In conclusion, we have shown that possible stable solitonswith ultraslow group velocity can be generated in a lifetime-broadened five-level hyper inverted-Y atomic system. Byanalysing the linear dynamics of the atomic system, we findthat there exist three regimes where the absorption of theweak pulsed probe field is almost completely suppressed whilesimultaneously the nonlinear dispersion effect is dramaticallyenhanced. These novel properties provide more preferences torealize ultraslow dark or bright optical solitons depending onthe three control fields’ intensities and the one-, two- and three-photon detunings. Therefore, the hyper inverted-Y atomicsystem described here represents schemes more flexible andadjustable in comparison with other schemes in atomic mediato form optical solitons and may lead to new phenomenathat manifest under well-controlled balance betweendispersion and nonlinear effects in an ultraslow propagationregime.

Acknowledgments

We would like to acknowledge Professor Y Wu for hisenlightening suggestions and encouragement. The workis supported in part by the NSF of China (Grant Nos60478029, 90503010, 10575040, 10874050, 10975054 and10634060), the National Fundamental Research Program ofChina (grant no 2005CB724508), and the Research Fund forthe Doctoral Program of Higher Education of China (grant no200804870051). Also, we acknowledge helpful advice andcomments from the referees.

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